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Sharp bounds for decomposing graphs into edges and triangles
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Blumenthal, Adam, Lidický, Bernard, Pehova, Yanitsa, Pfende, Florian, Pikhurko, Oleg and Volec, Jan (2021) Sharp bounds for decomposing graphs into edges and triangles. Combinatorics, Probability and Computing, 30 (2). 271 -287. doi:10.1017/S0963548320000358
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Official URL: http://dx.doi.org/10.1017/S0963548320000358
Abstract
For a real constant α, let π α 3 (G) be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let π α 3 (n) be the maximum of π α 3 (G) over all graphs G with n vertices. The extremal function π 3 3 (n) was first studied by Gy˝ori and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315– 320]. In a recent progress on this problem, Kr´al’, Lidick´y, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465–472] proved via flag algebras that π 3 3 (n) 6 (1/2+o(1))n 2 . We extend their result by determining the exact value of π α 3 (n) and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph K⌊n/2⌋,⌈n/2⌉ are the only possible extremal examples for large n.
| Item Type: | Journal Article | ||||||||||||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||||||||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||||||||||||||||||||
| Library of Congress Subject Headings (LCSH): | Decomposition (Mathematics), Graph theory, Triangle, Combinatorial analysis | ||||||||||||||||||||||||||||||
| Journal or Publication Title: | Combinatorics, Probability and Computing | ||||||||||||||||||||||||||||||
| Publisher: | Cambridge University Press | ||||||||||||||||||||||||||||||
| ISSN: | 0963-5483 | ||||||||||||||||||||||||||||||
| Official Date: | March 2021 | ||||||||||||||||||||||||||||||
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| Volume: | 30 | ||||||||||||||||||||||||||||||
| Number: | 2 | ||||||||||||||||||||||||||||||
| Page Range: | 271 -287 | ||||||||||||||||||||||||||||||
| DOI: | 10.1017/S0963548320000358 | ||||||||||||||||||||||||||||||
| Status: | Peer Reviewed | ||||||||||||||||||||||||||||||
| Publication Status: | Published | ||||||||||||||||||||||||||||||
| Access rights to Published version: | Open Access | ||||||||||||||||||||||||||||||
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